I have two separate and distinct problems with my computer that I hope you can help with! The first is as follows: A few weeks ago, a program called BearShare mysteriously appeared on my computer. I did not download it and had no idea where it came from or what it was so I uninstalled it and all the registry items that pertained to it. It is completely gone except when I try to use an existing web address in my address bar. No matter what address I select, BearShare comes up and I don't know how to get rid of it!
Problem #2: Whenver I come into this website, there is a yellow bar across the top of the page which reads as follows: "Your security settings do not allow websites to use ActiveX controls installed on you computer. This page may not display correctly. Click here for options...X." No matter what I do (close it or leave it open!), the bar reappears on every page of your website as I navigate through it.
I would appreciate any help I can get with these two issues. Thank you.

1/ Malware infection. Go get Malwarebytes and do a full scan.
2/ Are you using IE ? If so this seems to be an issue with a couple of the Google ads trying to use Flash. Hopefully point 1 should help there as well.

Geez Ben, no Firefox recommendation? :) That's a route I would have gone... plus a certain add-on. ;)

NK, I was just after confirmation first ;)

I ran Malwarebytes full scan and no threats were found and I still have both problems.

Check your hosts file.
Failing that try a system restore.

As for the second you didn't mention if you are running IE and which version.

I am running IE 7. What is the host file?

OK, well firstly upgrade to IE8 as it is infinity better and more secure.

Hosts file gives a IP to site name "shortcut".
It's located in Windows> System32> Drivers> etc folder.
Open with note pad

I have tried IE 8 and got rid of it. I had nothing but trouble with it.

That doesn't help you.
Hmm have you tried Firefox or Chrome, both miles better than IE ?

Yes anything but IE.